This paper deals with a multi-objective optimization criterion for linear viscous-elastic device utilised for decreasing vibrations induced in mechanical and structural systems by random loads. The proposed criterion for the optimum design is the minimization of a vector objective function. The multi-objective optimization is carried out by means of a stochastic approach. The design variables are the device frequency and the damping ratio. As cases of study, two different problems are analysed: the base isolation of a rigid mass and the tuned mass damper positioned on a multi degree of freedom structural system subject to a base acceleration. The non-dominated sorting genetic algorithm in its second version (NSGA-II) is adopted to obtain the Pareto sets and the corresponding optima for different characterizations of the system and input.